Combinatorial and Geometric Problems in Knot Theory

建议：DMS-9971244PI：Morwen Thilethwaite摘要：Threslethwaite的研究是在经典的纽结理论中，研究光滑的简单闭合曲线在三维球面上的嵌入。在他的研究中，计算机是必不可少的工具，因为他解决的问题往往是通过仔细观察数据来提出的。在他与C.Sundberg合作之后，Threslethwaite将继续研究节点数量的增长速度，在那里，确定了质数、交替链接的确切增长指数。他将研究对称纽结的最佳构型，并将研究与纽结补有关的其他几何问题，特别是包含本质上有4个穿孔的球体的双曲纽结补。他将调查双曲几何可能被用来为泰特飞行猜想提供一个新的、完全的几何证明的可能性。纽结表将远远超出目前列出的1,701,936节，这些新的表将用于搜索有趣的例子。免费提供的软件“纽结”将继续开发，它提供了一个图形化的纽结表界面和许多不变量的访问。希斯尔斯韦特的专长是经典纽结理论，起源于19世纪的三维拓扑学的一个分支。结是三维空间中的一条闭合曲线；可以想象一根打结的绳子，两端连接在一起。两个纽结被认为是等价的，如果一个纽结可以连续地变形到另一个，而一个纽结被认为是平凡的，或者说是未打结的，如果它等于一个平坦的圆。纽结理论是一门丰富的学科，因为它与几何学、拓扑学、代数和组合学相互作用。近年来，它引起了化学家和分子生物学家的兴趣，特别是与DNA打结有关的问题。纽结理论中有很多容易表述但难以解决的问题；例如，在实践中往往很难证明两个给定的纽结是不等价的，因为很难排除存在某种巧妙的将一个纽结变形为另一个纽结的方法。甚至很难判断一个特定的结是否微不足道。如果一个结被放置在具有尽可能少的交叉的平面上，则由此产生的交叉数称为该结的交叉数。到目前为止，希斯勒斯韦特已经用计算机对多达16个十字路口的1,701,936节进行了分类；这一分类得到了J.Hoste和J.Weks进行的独立制表的确认。作为该项目的一部分，他将把表格扩展到17或18个交叉路口，从而有望找到许多具有令人兴奋的特性的新例子。他将继续研究基本问题，即节点数量相对于交叉数的增长速度有多快。希斯尔斯韦特将使用双曲几何，非欧几里得几何的一种形式，来研究结的隐藏对称性，并确定交替结的结构(如果一个结的排列方式可以使绳索在连续的交叉点交替地上下穿行，那么它就是交替的。)他将继续开发软件包“结景”，它提供了一个图形界面的结表。

Proposal: DMS-9971244PI: Morwen ThistlethwaiteAbstract: Thistlethwaite's research is in classical knot theory, the study of embeddings of smooth simple closed curves in the 3-sphere. The computer is an essential tool in his research, in that the problems he addresses are often suggested by careful observation of data. Thistlethwaite will continue to investigate the rate of growth of the number of knots, following his collaboration with C. Sundberg, where the exact growth exponent was determined for prime, alternating links. He will study optimal configurations of symmetric knots, and will investigate other geometric problems associated with knot complements, in particular hyperbolic knot complements containing essential4-punctured spheres. He will investigate the possibility that hyperbolic geometry might be used to provide a new, completely geometric proof of the Tait flyping conjecture. The knot tables will be extended well beyond the 1,701,936 knots currently listed, and these new tables will be used to search for interesting examples. Development will continue on the freely available software package "Knotscape", which provides a graphical interface to the knot tables and access to many invariants.Thistlethwaite's specialty is classical knot theory, a branch of 3-dimensional topology with origins in the nineteenth century. A knot is a closed curve in 3-dimensional space; one can imagine a knotted rope with its ends joined together. Two knots are considered to be equivalent if one can be deformed continuously to the other, and a knot is deemed to be trivial, or unknotted, if it is equivalent to a flat circle. Knot theory is a rich subject, as it interfaces with geometry, topology, algebra and combinatorics. In recent years it has aroused the interest of chemists and molecular biologists, particularlyin relation to the knotting of DNA. Knot theory abounds with problems which are simple to state, but hard to solve; for example, it is often difficult in practice to prove that two given knots are inequivalent, as it is hard to rule out the existence of some ingenious method of deforming one to the other. It can even be hard to decide whether a given knot is trivial. If a knot is laid down on a flat surface with as few crossovers as possible, the resulting number of crossovers is called the crossing-number of the knot. To date, Thistlethwaite has classified by computer the 1,701,936 knots of up to 16 crossings; this classification was confirmed by an independent tabulation carried out by J. Hoste and J. Weeks. As part of this project, he will extend the tables to 17 or 18 crossings, thereby expecting to find many new examples with exciting properties. He will continue to investigate the fundamental problem as to how fast the number of knots grows in relation to crossing-number. Thistlethwaite will use hyperbolic geometry, a form of non-Euclidean geometry, to investigate hidden symmetries of knots, and to determine the structure of alternating knots (a knot is alternating if it can be arranged so that the rope goes alternately over and under at successive crossings.) He will continue to develop the software package "Knotscape", which provides a graphical interface to the knot tables.